Step |
Hyp |
Ref |
Expression |
1 |
|
signsply0.d |
|- D = ( deg ` F ) |
2 |
|
signsply0.c |
|- C = ( coeff ` F ) |
3 |
|
signsply0.b |
|- B = ( C ` D ) |
4 |
|
signsply0.a |
|- A = ( C ` 0 ) |
5 |
|
signsply0.1 |
|- ( ph -> F e. ( Poly ` RR ) ) |
6 |
|
signsply0.2 |
|- ( ph -> F =/= 0p ) |
7 |
|
signsply0.3 |
|- ( ph -> ( A x. B ) < 0 ) |
8 |
|
simplr |
|- ( ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) /\ A. f e. RR+ ( d <_ f -> ( abs ` ( ( ( F ` f ) / ( f ^ D ) ) - B ) ) < -u B ) ) -> d e. RR+ ) |
9 |
|
simpr |
|- ( ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) /\ A. f e. RR+ ( d <_ f -> ( abs ` ( ( ( F ` f ) / ( f ^ D ) ) - B ) ) < -u B ) ) -> A. f e. RR+ ( d <_ f -> ( abs ` ( ( ( F ` f ) / ( f ^ D ) ) - B ) ) < -u B ) ) |
10 |
|
rpxr |
|- ( d e. RR+ -> d e. RR* ) |
11 |
10
|
xrleidd |
|- ( d e. RR+ -> d <_ d ) |
12 |
11
|
ad2antlr |
|- ( ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) /\ A. f e. RR+ ( d <_ f -> ( abs ` ( ( ( F ` f ) / ( f ^ D ) ) - B ) ) < -u B ) ) -> d <_ d ) |
13 |
|
id |
|- ( d e. RR+ -> d e. RR+ ) |
14 |
|
simpr |
|- ( ( d e. RR+ /\ f = d ) -> f = d ) |
15 |
14
|
breq2d |
|- ( ( d e. RR+ /\ f = d ) -> ( d <_ f <-> d <_ d ) ) |
16 |
14
|
fveq2d |
|- ( ( d e. RR+ /\ f = d ) -> ( F ` f ) = ( F ` d ) ) |
17 |
14
|
oveq1d |
|- ( ( d e. RR+ /\ f = d ) -> ( f ^ D ) = ( d ^ D ) ) |
18 |
16 17
|
oveq12d |
|- ( ( d e. RR+ /\ f = d ) -> ( ( F ` f ) / ( f ^ D ) ) = ( ( F ` d ) / ( d ^ D ) ) ) |
19 |
18
|
fvoveq1d |
|- ( ( d e. RR+ /\ f = d ) -> ( abs ` ( ( ( F ` f ) / ( f ^ D ) ) - B ) ) = ( abs ` ( ( ( F ` d ) / ( d ^ D ) ) - B ) ) ) |
20 |
19
|
breq1d |
|- ( ( d e. RR+ /\ f = d ) -> ( ( abs ` ( ( ( F ` f ) / ( f ^ D ) ) - B ) ) < -u B <-> ( abs ` ( ( ( F ` d ) / ( d ^ D ) ) - B ) ) < -u B ) ) |
21 |
15 20
|
imbi12d |
|- ( ( d e. RR+ /\ f = d ) -> ( ( d <_ f -> ( abs ` ( ( ( F ` f ) / ( f ^ D ) ) - B ) ) < -u B ) <-> ( d <_ d -> ( abs ` ( ( ( F ` d ) / ( d ^ D ) ) - B ) ) < -u B ) ) ) |
22 |
13 21
|
rspcdv |
|- ( d e. RR+ -> ( A. f e. RR+ ( d <_ f -> ( abs ` ( ( ( F ` f ) / ( f ^ D ) ) - B ) ) < -u B ) -> ( d <_ d -> ( abs ` ( ( ( F ` d ) / ( d ^ D ) ) - B ) ) < -u B ) ) ) |
23 |
8 9 12 22
|
syl3c |
|- ( ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) /\ A. f e. RR+ ( d <_ f -> ( abs ` ( ( ( F ` f ) / ( f ^ D ) ) - B ) ) < -u B ) ) -> ( abs ` ( ( ( F ` d ) / ( d ^ D ) ) - B ) ) < -u B ) |
24 |
5
|
ad2antrr |
|- ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) -> F e. ( Poly ` RR ) ) |
25 |
|
simpr |
|- ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) -> d e. RR+ ) |
26 |
25
|
rpred |
|- ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) -> d e. RR ) |
27 |
24 26
|
plyrecld |
|- ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) -> ( F ` d ) e. RR ) |
28 |
|
dgrcl |
|- ( F e. ( Poly ` RR ) -> ( deg ` F ) e. NN0 ) |
29 |
5 28
|
syl |
|- ( ph -> ( deg ` F ) e. NN0 ) |
30 |
1 29
|
eqeltrid |
|- ( ph -> D e. NN0 ) |
31 |
30
|
ad2antrr |
|- ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) -> D e. NN0 ) |
32 |
26 31
|
reexpcld |
|- ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) -> ( d ^ D ) e. RR ) |
33 |
25
|
rpcnd |
|- ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) -> d e. CC ) |
34 |
25
|
rpne0d |
|- ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) -> d =/= 0 ) |
35 |
30
|
nn0zd |
|- ( ph -> D e. ZZ ) |
36 |
35
|
ad2antrr |
|- ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) -> D e. ZZ ) |
37 |
33 34 36
|
expne0d |
|- ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) -> ( d ^ D ) =/= 0 ) |
38 |
27 32 37
|
redivcld |
|- ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) -> ( ( F ` d ) / ( d ^ D ) ) e. RR ) |
39 |
|
0re |
|- 0 e. RR |
40 |
2
|
coef2 |
|- ( ( F e. ( Poly ` RR ) /\ 0 e. RR ) -> C : NN0 --> RR ) |
41 |
39 40
|
mpan2 |
|- ( F e. ( Poly ` RR ) -> C : NN0 --> RR ) |
42 |
41
|
ffvelrnda |
|- ( ( F e. ( Poly ` RR ) /\ D e. NN0 ) -> ( C ` D ) e. RR ) |
43 |
3 42
|
eqeltrid |
|- ( ( F e. ( Poly ` RR ) /\ D e. NN0 ) -> B e. RR ) |
44 |
5 30 43
|
syl2anc |
|- ( ph -> B e. RR ) |
45 |
44
|
ad2antrr |
|- ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) -> B e. RR ) |
46 |
45
|
renegcld |
|- ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) -> -u B e. RR ) |
47 |
38 45 46
|
absdifltd |
|- ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) -> ( ( abs ` ( ( ( F ` d ) / ( d ^ D ) ) - B ) ) < -u B <-> ( ( B - -u B ) < ( ( F ` d ) / ( d ^ D ) ) /\ ( ( F ` d ) / ( d ^ D ) ) < ( B + -u B ) ) ) ) |
48 |
47
|
simplbda |
|- ( ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) /\ ( abs ` ( ( ( F ` d ) / ( d ^ D ) ) - B ) ) < -u B ) -> ( ( F ` d ) / ( d ^ D ) ) < ( B + -u B ) ) |
49 |
44
|
recnd |
|- ( ph -> B e. CC ) |
50 |
49
|
ad2antrr |
|- ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) -> B e. CC ) |
51 |
50
|
negidd |
|- ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) -> ( B + -u B ) = 0 ) |
52 |
51
|
adantr |
|- ( ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) /\ ( abs ` ( ( ( F ` d ) / ( d ^ D ) ) - B ) ) < -u B ) -> ( B + -u B ) = 0 ) |
53 |
48 52
|
breqtrd |
|- ( ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) /\ ( abs ` ( ( ( F ` d ) / ( d ^ D ) ) - B ) ) < -u B ) -> ( ( F ` d ) / ( d ^ D ) ) < 0 ) |
54 |
25 36
|
rpexpcld |
|- ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) -> ( d ^ D ) e. RR+ ) |
55 |
27 54
|
ge0divd |
|- ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) -> ( 0 <_ ( F ` d ) <-> 0 <_ ( ( F ` d ) / ( d ^ D ) ) ) ) |
56 |
55
|
notbid |
|- ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) -> ( -. 0 <_ ( F ` d ) <-> -. 0 <_ ( ( F ` d ) / ( d ^ D ) ) ) ) |
57 |
|
0red |
|- ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) -> 0 e. RR ) |
58 |
27 57
|
ltnled |
|- ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) -> ( ( F ` d ) < 0 <-> -. 0 <_ ( F ` d ) ) ) |
59 |
38 57
|
ltnled |
|- ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) -> ( ( ( F ` d ) / ( d ^ D ) ) < 0 <-> -. 0 <_ ( ( F ` d ) / ( d ^ D ) ) ) ) |
60 |
56 58 59
|
3bitr4d |
|- ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) -> ( ( F ` d ) < 0 <-> ( ( F ` d ) / ( d ^ D ) ) < 0 ) ) |
61 |
60
|
adantr |
|- ( ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) /\ ( abs ` ( ( ( F ` d ) / ( d ^ D ) ) - B ) ) < -u B ) -> ( ( F ` d ) < 0 <-> ( ( F ` d ) / ( d ^ D ) ) < 0 ) ) |
62 |
53 61
|
mpbird |
|- ( ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) /\ ( abs ` ( ( ( F ` d ) / ( d ^ D ) ) - B ) ) < -u B ) -> ( F ` d ) < 0 ) |
63 |
23 62
|
syldan |
|- ( ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) /\ A. f e. RR+ ( d <_ f -> ( abs ` ( ( ( F ` f ) / ( f ^ D ) ) - B ) ) < -u B ) ) -> ( F ` d ) < 0 ) |
64 |
|
0red |
|- ( ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) /\ ( F ` d ) < 0 ) -> 0 e. RR ) |
65 |
|
simplr |
|- ( ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) /\ ( F ` d ) < 0 ) -> d e. RR+ ) |
66 |
65
|
rpred |
|- ( ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) /\ ( F ` d ) < 0 ) -> d e. RR ) |
67 |
65
|
rpgt0d |
|- ( ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) /\ ( F ` d ) < 0 ) -> 0 < d ) |
68 |
|
iccssre |
|- ( ( 0 e. RR /\ d e. RR ) -> ( 0 [,] d ) C_ RR ) |
69 |
39 66 68
|
sylancr |
|- ( ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) /\ ( F ` d ) < 0 ) -> ( 0 [,] d ) C_ RR ) |
70 |
|
ax-resscn |
|- RR C_ CC |
71 |
69 70
|
sstrdi |
|- ( ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) /\ ( F ` d ) < 0 ) -> ( 0 [,] d ) C_ CC ) |
72 |
|
plycn |
|- ( F e. ( Poly ` RR ) -> F e. ( CC -cn-> CC ) ) |
73 |
5 72
|
syl |
|- ( ph -> F e. ( CC -cn-> CC ) ) |
74 |
73
|
ad3antrrr |
|- ( ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) /\ ( F ` d ) < 0 ) -> F e. ( CC -cn-> CC ) ) |
75 |
5
|
ad4antr |
|- ( ( ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) /\ ( F ` d ) < 0 ) /\ x e. ( 0 [,] d ) ) -> F e. ( Poly ` RR ) ) |
76 |
69
|
sselda |
|- ( ( ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) /\ ( F ` d ) < 0 ) /\ x e. ( 0 [,] d ) ) -> x e. RR ) |
77 |
75 76
|
plyrecld |
|- ( ( ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) /\ ( F ` d ) < 0 ) /\ x e. ( 0 [,] d ) ) -> ( F ` x ) e. RR ) |
78 |
|
simpr |
|- ( ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) /\ ( F ` d ) < 0 ) -> ( F ` d ) < 0 ) |
79 |
|
simplll |
|- ( ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) /\ ( F ` d ) < 0 ) -> ph ) |
80 |
79 44
|
syl |
|- ( ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) /\ ( F ` d ) < 0 ) -> B e. RR ) |
81 |
|
simpr |
|- ( ( ph /\ -u B e. RR+ ) -> -u B e. RR+ ) |
82 |
81
|
ad2antrr |
|- ( ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) /\ ( F ` d ) < 0 ) -> -u B e. RR+ ) |
83 |
|
negelrp |
|- ( B e. RR -> ( -u B e. RR+ <-> B < 0 ) ) |
84 |
83
|
biimpa |
|- ( ( B e. RR /\ -u B e. RR+ ) -> B < 0 ) |
85 |
80 82 84
|
syl2anc |
|- ( ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) /\ ( F ` d ) < 0 ) -> B < 0 ) |
86 |
5 39 40
|
sylancl |
|- ( ph -> C : NN0 --> RR ) |
87 |
|
0nn0 |
|- 0 e. NN0 |
88 |
87
|
a1i |
|- ( ph -> 0 e. NN0 ) |
89 |
86 88
|
ffvelrnd |
|- ( ph -> ( C ` 0 ) e. RR ) |
90 |
4 89
|
eqeltrid |
|- ( ph -> A e. RR ) |
91 |
90 44 7
|
mul2lt0rlt0 |
|- ( ( ph /\ B < 0 ) -> 0 < A ) |
92 |
91 4
|
breqtrdi |
|- ( ( ph /\ B < 0 ) -> 0 < ( C ` 0 ) ) |
93 |
79 85 92
|
syl2anc |
|- ( ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) /\ ( F ` d ) < 0 ) -> 0 < ( C ` 0 ) ) |
94 |
2
|
coefv0 |
|- ( F e. ( Poly ` RR ) -> ( F ` 0 ) = ( C ` 0 ) ) |
95 |
5 94
|
syl |
|- ( ph -> ( F ` 0 ) = ( C ` 0 ) ) |
96 |
95
|
ad3antrrr |
|- ( ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) /\ ( F ` d ) < 0 ) -> ( F ` 0 ) = ( C ` 0 ) ) |
97 |
93 96
|
breqtrrd |
|- ( ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) /\ ( F ` d ) < 0 ) -> 0 < ( F ` 0 ) ) |
98 |
78 97
|
jca |
|- ( ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) /\ ( F ` d ) < 0 ) -> ( ( F ` d ) < 0 /\ 0 < ( F ` 0 ) ) ) |
99 |
64 66 64 67 71 74 77 98
|
ivth2 |
|- ( ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) /\ ( F ` d ) < 0 ) -> E. z e. ( 0 (,) d ) ( F ` z ) = 0 ) |
100 |
|
0le0 |
|- 0 <_ 0 |
101 |
|
pnfge |
|- ( d e. RR* -> d <_ +oo ) |
102 |
10 101
|
syl |
|- ( d e. RR+ -> d <_ +oo ) |
103 |
|
0xr |
|- 0 e. RR* |
104 |
|
pnfxr |
|- +oo e. RR* |
105 |
|
ioossioo |
|- ( ( ( 0 e. RR* /\ +oo e. RR* ) /\ ( 0 <_ 0 /\ d <_ +oo ) ) -> ( 0 (,) d ) C_ ( 0 (,) +oo ) ) |
106 |
103 104 105
|
mpanl12 |
|- ( ( 0 <_ 0 /\ d <_ +oo ) -> ( 0 (,) d ) C_ ( 0 (,) +oo ) ) |
107 |
100 102 106
|
sylancr |
|- ( d e. RR+ -> ( 0 (,) d ) C_ ( 0 (,) +oo ) ) |
108 |
|
ioorp |
|- ( 0 (,) +oo ) = RR+ |
109 |
107 108
|
sseqtrdi |
|- ( d e. RR+ -> ( 0 (,) d ) C_ RR+ ) |
110 |
|
ssrexv |
|- ( ( 0 (,) d ) C_ RR+ -> ( E. z e. ( 0 (,) d ) ( F ` z ) = 0 -> E. z e. RR+ ( F ` z ) = 0 ) ) |
111 |
65 109 110
|
3syl |
|- ( ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) /\ ( F ` d ) < 0 ) -> ( E. z e. ( 0 (,) d ) ( F ` z ) = 0 -> E. z e. RR+ ( F ` z ) = 0 ) ) |
112 |
99 111
|
mpd |
|- ( ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) /\ ( F ` d ) < 0 ) -> E. z e. RR+ ( F ` z ) = 0 ) |
113 |
63 112
|
syldan |
|- ( ( ( ( ph /\ -u B e. RR+ ) /\ d e. RR+ ) /\ A. f e. RR+ ( d <_ f -> ( abs ` ( ( ( F ` f ) / ( f ^ D ) ) - B ) ) < -u B ) ) -> E. z e. RR+ ( F ` z ) = 0 ) |
114 |
|
plyf |
|- ( F e. ( Poly ` RR ) -> F : CC --> CC ) |
115 |
5 114
|
syl |
|- ( ph -> F : CC --> CC ) |
116 |
115
|
ffnd |
|- ( ph -> F Fn CC ) |
117 |
|
ovex |
|- ( x ^ D ) e. _V |
118 |
117
|
rgenw |
|- A. x e. RR+ ( x ^ D ) e. _V |
119 |
|
eqid |
|- ( x e. RR+ |-> ( x ^ D ) ) = ( x e. RR+ |-> ( x ^ D ) ) |
120 |
119
|
fnmpt |
|- ( A. x e. RR+ ( x ^ D ) e. _V -> ( x e. RR+ |-> ( x ^ D ) ) Fn RR+ ) |
121 |
118 120
|
mp1i |
|- ( ph -> ( x e. RR+ |-> ( x ^ D ) ) Fn RR+ ) |
122 |
|
cnex |
|- CC e. _V |
123 |
122
|
a1i |
|- ( ph -> CC e. _V ) |
124 |
|
rpssre |
|- RR+ C_ RR |
125 |
124 70
|
sstri |
|- RR+ C_ CC |
126 |
122 125
|
ssexi |
|- RR+ e. _V |
127 |
126
|
a1i |
|- ( ph -> RR+ e. _V ) |
128 |
|
sseqin2 |
|- ( RR+ C_ CC <-> ( CC i^i RR+ ) = RR+ ) |
129 |
125 128
|
mpbi |
|- ( CC i^i RR+ ) = RR+ |
130 |
|
eqidd |
|- ( ( ph /\ f e. CC ) -> ( F ` f ) = ( F ` f ) ) |
131 |
|
eqidd |
|- ( ( ph /\ f e. RR+ ) -> ( x e. RR+ |-> ( x ^ D ) ) = ( x e. RR+ |-> ( x ^ D ) ) ) |
132 |
|
simpr |
|- ( ( ( ph /\ f e. RR+ ) /\ x = f ) -> x = f ) |
133 |
132
|
oveq1d |
|- ( ( ( ph /\ f e. RR+ ) /\ x = f ) -> ( x ^ D ) = ( f ^ D ) ) |
134 |
|
simpr |
|- ( ( ph /\ f e. RR+ ) -> f e. RR+ ) |
135 |
|
ovexd |
|- ( ( ph /\ f e. RR+ ) -> ( f ^ D ) e. _V ) |
136 |
131 133 134 135
|
fvmptd |
|- ( ( ph /\ f e. RR+ ) -> ( ( x e. RR+ |-> ( x ^ D ) ) ` f ) = ( f ^ D ) ) |
137 |
116 121 123 127 129 130 136
|
offval |
|- ( ph -> ( F oF / ( x e. RR+ |-> ( x ^ D ) ) ) = ( f e. RR+ |-> ( ( F ` f ) / ( f ^ D ) ) ) ) |
138 |
|
oveq1 |
|- ( x = f -> ( x ^ D ) = ( f ^ D ) ) |
139 |
138
|
cbvmptv |
|- ( x e. RR+ |-> ( x ^ D ) ) = ( f e. RR+ |-> ( f ^ D ) ) |
140 |
1 2 3 139
|
signsplypnf |
|- ( F e. ( Poly ` RR ) -> ( F oF / ( x e. RR+ |-> ( x ^ D ) ) ) ~~>r B ) |
141 |
5 140
|
syl |
|- ( ph -> ( F oF / ( x e. RR+ |-> ( x ^ D ) ) ) ~~>r B ) |
142 |
137 141
|
eqbrtrrd |
|- ( ph -> ( f e. RR+ |-> ( ( F ` f ) / ( f ^ D ) ) ) ~~>r B ) |
143 |
115
|
adantr |
|- ( ( ph /\ f e. RR+ ) -> F : CC --> CC ) |
144 |
134
|
rpcnd |
|- ( ( ph /\ f e. RR+ ) -> f e. CC ) |
145 |
143 144
|
ffvelrnd |
|- ( ( ph /\ f e. RR+ ) -> ( F ` f ) e. CC ) |
146 |
30
|
adantr |
|- ( ( ph /\ f e. RR+ ) -> D e. NN0 ) |
147 |
144 146
|
expcld |
|- ( ( ph /\ f e. RR+ ) -> ( f ^ D ) e. CC ) |
148 |
134
|
rpne0d |
|- ( ( ph /\ f e. RR+ ) -> f =/= 0 ) |
149 |
35
|
adantr |
|- ( ( ph /\ f e. RR+ ) -> D e. ZZ ) |
150 |
144 148 149
|
expne0d |
|- ( ( ph /\ f e. RR+ ) -> ( f ^ D ) =/= 0 ) |
151 |
145 147 150
|
divcld |
|- ( ( ph /\ f e. RR+ ) -> ( ( F ` f ) / ( f ^ D ) ) e. CC ) |
152 |
151
|
ralrimiva |
|- ( ph -> A. f e. RR+ ( ( F ` f ) / ( f ^ D ) ) e. CC ) |
153 |
124
|
a1i |
|- ( ph -> RR+ C_ RR ) |
154 |
|
1red |
|- ( ph -> 1 e. RR ) |
155 |
152 153 49 154
|
rlim3 |
|- ( ph -> ( ( f e. RR+ |-> ( ( F ` f ) / ( f ^ D ) ) ) ~~>r B <-> A. e e. RR+ E. d e. ( 1 [,) +oo ) A. f e. RR+ ( d <_ f -> ( abs ` ( ( ( F ` f ) / ( f ^ D ) ) - B ) ) < e ) ) ) |
156 |
142 155
|
mpbid |
|- ( ph -> A. e e. RR+ E. d e. ( 1 [,) +oo ) A. f e. RR+ ( d <_ f -> ( abs ` ( ( ( F ` f ) / ( f ^ D ) ) - B ) ) < e ) ) |
157 |
|
0lt1 |
|- 0 < 1 |
158 |
|
pnfge |
|- ( +oo e. RR* -> +oo <_ +oo ) |
159 |
104 158
|
ax-mp |
|- +oo <_ +oo |
160 |
|
icossioo |
|- ( ( ( 0 e. RR* /\ +oo e. RR* ) /\ ( 0 < 1 /\ +oo <_ +oo ) ) -> ( 1 [,) +oo ) C_ ( 0 (,) +oo ) ) |
161 |
103 104 157 159 160
|
mp4an |
|- ( 1 [,) +oo ) C_ ( 0 (,) +oo ) |
162 |
161 108
|
sseqtri |
|- ( 1 [,) +oo ) C_ RR+ |
163 |
|
ssrexv |
|- ( ( 1 [,) +oo ) C_ RR+ -> ( E. d e. ( 1 [,) +oo ) A. f e. RR+ ( d <_ f -> ( abs ` ( ( ( F ` f ) / ( f ^ D ) ) - B ) ) < e ) -> E. d e. RR+ A. f e. RR+ ( d <_ f -> ( abs ` ( ( ( F ` f ) / ( f ^ D ) ) - B ) ) < e ) ) ) |
164 |
162 163
|
ax-mp |
|- ( E. d e. ( 1 [,) +oo ) A. f e. RR+ ( d <_ f -> ( abs ` ( ( ( F ` f ) / ( f ^ D ) ) - B ) ) < e ) -> E. d e. RR+ A. f e. RR+ ( d <_ f -> ( abs ` ( ( ( F ` f ) / ( f ^ D ) ) - B ) ) < e ) ) |
165 |
164
|
ralimi |
|- ( A. e e. RR+ E. d e. ( 1 [,) +oo ) A. f e. RR+ ( d <_ f -> ( abs ` ( ( ( F ` f ) / ( f ^ D ) ) - B ) ) < e ) -> A. e e. RR+ E. d e. RR+ A. f e. RR+ ( d <_ f -> ( abs ` ( ( ( F ` f ) / ( f ^ D ) ) - B ) ) < e ) ) |
166 |
156 165
|
syl |
|- ( ph -> A. e e. RR+ E. d e. RR+ A. f e. RR+ ( d <_ f -> ( abs ` ( ( ( F ` f ) / ( f ^ D ) ) - B ) ) < e ) ) |
167 |
166
|
adantr |
|- ( ( ph /\ -u B e. RR+ ) -> A. e e. RR+ E. d e. RR+ A. f e. RR+ ( d <_ f -> ( abs ` ( ( ( F ` f ) / ( f ^ D ) ) - B ) ) < e ) ) |
168 |
|
simpr |
|- ( ( ( ph /\ -u B e. RR+ ) /\ e = -u B ) -> e = -u B ) |
169 |
168
|
breq2d |
|- ( ( ( ph /\ -u B e. RR+ ) /\ e = -u B ) -> ( ( abs ` ( ( ( F ` f ) / ( f ^ D ) ) - B ) ) < e <-> ( abs ` ( ( ( F ` f ) / ( f ^ D ) ) - B ) ) < -u B ) ) |
170 |
169
|
imbi2d |
|- ( ( ( ph /\ -u B e. RR+ ) /\ e = -u B ) -> ( ( d <_ f -> ( abs ` ( ( ( F ` f ) / ( f ^ D ) ) - B ) ) < e ) <-> ( d <_ f -> ( abs ` ( ( ( F ` f ) / ( f ^ D ) ) - B ) ) < -u B ) ) ) |
171 |
170
|
rexralbidv |
|- ( ( ( ph /\ -u B e. RR+ ) /\ e = -u B ) -> ( E. d e. RR+ A. f e. RR+ ( d <_ f -> ( abs ` ( ( ( F ` f ) / ( f ^ D ) ) - B ) ) < e ) <-> E. d e. RR+ A. f e. RR+ ( d <_ f -> ( abs ` ( ( ( F ` f ) / ( f ^ D ) ) - B ) ) < -u B ) ) ) |
172 |
81 171
|
rspcdv |
|- ( ( ph /\ -u B e. RR+ ) -> ( A. e e. RR+ E. d e. RR+ A. f e. RR+ ( d <_ f -> ( abs ` ( ( ( F ` f ) / ( f ^ D ) ) - B ) ) < e ) -> E. d e. RR+ A. f e. RR+ ( d <_ f -> ( abs ` ( ( ( F ` f ) / ( f ^ D ) ) - B ) ) < -u B ) ) ) |
173 |
167 172
|
mpd |
|- ( ( ph /\ -u B e. RR+ ) -> E. d e. RR+ A. f e. RR+ ( d <_ f -> ( abs ` ( ( ( F ` f ) / ( f ^ D ) ) - B ) ) < -u B ) ) |
174 |
113 173
|
r19.29a |
|- ( ( ph /\ -u B e. RR+ ) -> E. z e. RR+ ( F ` z ) = 0 ) |
175 |
|
simplr |
|- ( ( ( ( ph /\ B e. RR+ ) /\ d e. RR+ ) /\ A. f e. RR+ ( d <_ f -> ( abs ` ( ( ( F ` f ) / ( f ^ D ) ) - B ) ) < B ) ) -> d e. RR+ ) |
176 |
|
simpr |
|- ( ( ( ( ph /\ B e. RR+ ) /\ d e. RR+ ) /\ A. f e. RR+ ( d <_ f -> ( abs ` ( ( ( F ` f ) / ( f ^ D ) ) - B ) ) < B ) ) -> A. f e. RR+ ( d <_ f -> ( abs ` ( ( ( F ` f ) / ( f ^ D ) ) - B ) ) < B ) ) |
177 |
11
|
ad2antlr |
|- ( ( ( ( ph /\ B e. RR+ ) /\ d e. RR+ ) /\ A. f e. RR+ ( d <_ f -> ( abs ` ( ( ( F ` f ) / ( f ^ D ) ) - B ) ) < B ) ) -> d <_ d ) |
178 |
19
|
breq1d |
|- ( ( d e. RR+ /\ f = d ) -> ( ( abs ` ( ( ( F ` f ) / ( f ^ D ) ) - B ) ) < B <-> ( abs ` ( ( ( F ` d ) / ( d ^ D ) ) - B ) ) < B ) ) |
179 |
15 178
|
imbi12d |
|- ( ( d e. RR+ /\ f = d ) -> ( ( d <_ f -> ( abs ` ( ( ( F ` f ) / ( f ^ D ) ) - B ) ) < B ) <-> ( d <_ d -> ( abs ` ( ( ( F ` d ) / ( d ^ D ) ) - B ) ) < B ) ) ) |
180 |
13 179
|
rspcdv |
|- ( d e. RR+ -> ( A. f e. RR+ ( d <_ f -> ( abs ` ( ( ( F ` f ) / ( f ^ D ) ) - B ) ) < B ) -> ( d <_ d -> ( abs ` ( ( ( F ` d ) / ( d ^ D ) ) - B ) ) < B ) ) ) |
181 |
175 176 177 180
|
syl3c |
|- ( ( ( ( ph /\ B e. RR+ ) /\ d e. RR+ ) /\ A. f e. RR+ ( d <_ f -> ( abs ` ( ( ( F ` f ) / ( f ^ D ) ) - B ) ) < B ) ) -> ( abs ` ( ( ( F ` d ) / ( d ^ D ) ) - B ) ) < B ) |
182 |
49
|
ad2antrr |
|- ( ( ( ph /\ B e. RR+ ) /\ d e. RR+ ) -> B e. CC ) |
183 |
182
|
subidd |
|- ( ( ( ph /\ B e. RR+ ) /\ d e. RR+ ) -> ( B - B ) = 0 ) |
184 |
183
|
adantr |
|- ( ( ( ( ph /\ B e. RR+ ) /\ d e. RR+ ) /\ ( abs ` ( ( ( F ` d ) / ( d ^ D ) ) - B ) ) < B ) -> ( B - B ) = 0 ) |
185 |
5
|
ad2antrr |
|- ( ( ( ph /\ B e. RR+ ) /\ d e. RR+ ) -> F e. ( Poly ` RR ) ) |
186 |
124
|
a1i |
|- ( ( ph /\ B e. RR+ ) -> RR+ C_ RR ) |
187 |
186
|
sselda |
|- ( ( ( ph /\ B e. RR+ ) /\ d e. RR+ ) -> d e. RR ) |
188 |
185 187
|
plyrecld |
|- ( ( ( ph /\ B e. RR+ ) /\ d e. RR+ ) -> ( F ` d ) e. RR ) |
189 |
30
|
ad2antrr |
|- ( ( ( ph /\ B e. RR+ ) /\ d e. RR+ ) -> D e. NN0 ) |
190 |
187 189
|
reexpcld |
|- ( ( ( ph /\ B e. RR+ ) /\ d e. RR+ ) -> ( d ^ D ) e. RR ) |
191 |
187
|
recnd |
|- ( ( ( ph /\ B e. RR+ ) /\ d e. RR+ ) -> d e. CC ) |
192 |
|
simpr |
|- ( ( ( ph /\ B e. RR+ ) /\ d e. RR+ ) -> d e. RR+ ) |
193 |
192
|
rpne0d |
|- ( ( ( ph /\ B e. RR+ ) /\ d e. RR+ ) -> d =/= 0 ) |
194 |
35
|
ad2antrr |
|- ( ( ( ph /\ B e. RR+ ) /\ d e. RR+ ) -> D e. ZZ ) |
195 |
191 193 194
|
expne0d |
|- ( ( ( ph /\ B e. RR+ ) /\ d e. RR+ ) -> ( d ^ D ) =/= 0 ) |
196 |
188 190 195
|
redivcld |
|- ( ( ( ph /\ B e. RR+ ) /\ d e. RR+ ) -> ( ( F ` d ) / ( d ^ D ) ) e. RR ) |
197 |
44
|
ad2antrr |
|- ( ( ( ph /\ B e. RR+ ) /\ d e. RR+ ) -> B e. RR ) |
198 |
196 197 197
|
absdifltd |
|- ( ( ( ph /\ B e. RR+ ) /\ d e. RR+ ) -> ( ( abs ` ( ( ( F ` d ) / ( d ^ D ) ) - B ) ) < B <-> ( ( B - B ) < ( ( F ` d ) / ( d ^ D ) ) /\ ( ( F ` d ) / ( d ^ D ) ) < ( B + B ) ) ) ) |
199 |
198
|
simprbda |
|- ( ( ( ( ph /\ B e. RR+ ) /\ d e. RR+ ) /\ ( abs ` ( ( ( F ` d ) / ( d ^ D ) ) - B ) ) < B ) -> ( B - B ) < ( ( F ` d ) / ( d ^ D ) ) ) |
200 |
184 199
|
eqbrtrrd |
|- ( ( ( ( ph /\ B e. RR+ ) /\ d e. RR+ ) /\ ( abs ` ( ( ( F ` d ) / ( d ^ D ) ) - B ) ) < B ) -> 0 < ( ( F ` d ) / ( d ^ D ) ) ) |
201 |
192 194
|
rpexpcld |
|- ( ( ( ph /\ B e. RR+ ) /\ d e. RR+ ) -> ( d ^ D ) e. RR+ ) |
202 |
188 201
|
gt0divd |
|- ( ( ( ph /\ B e. RR+ ) /\ d e. RR+ ) -> ( 0 < ( F ` d ) <-> 0 < ( ( F ` d ) / ( d ^ D ) ) ) ) |
203 |
202
|
adantr |
|- ( ( ( ( ph /\ B e. RR+ ) /\ d e. RR+ ) /\ ( abs ` ( ( ( F ` d ) / ( d ^ D ) ) - B ) ) < B ) -> ( 0 < ( F ` d ) <-> 0 < ( ( F ` d ) / ( d ^ D ) ) ) ) |
204 |
200 203
|
mpbird |
|- ( ( ( ( ph /\ B e. RR+ ) /\ d e. RR+ ) /\ ( abs ` ( ( ( F ` d ) / ( d ^ D ) ) - B ) ) < B ) -> 0 < ( F ` d ) ) |
205 |
181 204
|
syldan |
|- ( ( ( ( ph /\ B e. RR+ ) /\ d e. RR+ ) /\ A. f e. RR+ ( d <_ f -> ( abs ` ( ( ( F ` f ) / ( f ^ D ) ) - B ) ) < B ) ) -> 0 < ( F ` d ) ) |
206 |
|
0red |
|- ( ( ( ( ph /\ B e. RR+ ) /\ d e. RR+ ) /\ 0 < ( F ` d ) ) -> 0 e. RR ) |
207 |
|
simplr |
|- ( ( ( ( ph /\ B e. RR+ ) /\ d e. RR+ ) /\ 0 < ( F ` d ) ) -> d e. RR+ ) |
208 |
207
|
rpred |
|- ( ( ( ( ph /\ B e. RR+ ) /\ d e. RR+ ) /\ 0 < ( F ` d ) ) -> d e. RR ) |
209 |
207
|
rpgt0d |
|- ( ( ( ( ph /\ B e. RR+ ) /\ d e. RR+ ) /\ 0 < ( F ` d ) ) -> 0 < d ) |
210 |
39 208 68
|
sylancr |
|- ( ( ( ( ph /\ B e. RR+ ) /\ d e. RR+ ) /\ 0 < ( F ` d ) ) -> ( 0 [,] d ) C_ RR ) |
211 |
210 70
|
sstrdi |
|- ( ( ( ( ph /\ B e. RR+ ) /\ d e. RR+ ) /\ 0 < ( F ` d ) ) -> ( 0 [,] d ) C_ CC ) |
212 |
73
|
ad3antrrr |
|- ( ( ( ( ph /\ B e. RR+ ) /\ d e. RR+ ) /\ 0 < ( F ` d ) ) -> F e. ( CC -cn-> CC ) ) |
213 |
5
|
ad4antr |
|- ( ( ( ( ( ph /\ B e. RR+ ) /\ d e. RR+ ) /\ 0 < ( F ` d ) ) /\ x e. ( 0 [,] d ) ) -> F e. ( Poly ` RR ) ) |
214 |
210
|
sselda |
|- ( ( ( ( ( ph /\ B e. RR+ ) /\ d e. RR+ ) /\ 0 < ( F ` d ) ) /\ x e. ( 0 [,] d ) ) -> x e. RR ) |
215 |
213 214
|
plyrecld |
|- ( ( ( ( ( ph /\ B e. RR+ ) /\ d e. RR+ ) /\ 0 < ( F ` d ) ) /\ x e. ( 0 [,] d ) ) -> ( F ` x ) e. RR ) |
216 |
95
|
ad3antrrr |
|- ( ( ( ( ph /\ B e. RR+ ) /\ d e. RR+ ) /\ 0 < ( F ` d ) ) -> ( F ` 0 ) = ( C ` 0 ) ) |
217 |
|
simplll |
|- ( ( ( ( ph /\ B e. RR+ ) /\ d e. RR+ ) /\ 0 < ( F ` d ) ) -> ph ) |
218 |
|
simpr1 |
|- ( ( ph /\ ( B e. RR+ /\ d e. RR+ /\ 0 < ( F ` d ) ) ) -> B e. RR+ ) |
219 |
218
|
rpgt0d |
|- ( ( ph /\ ( B e. RR+ /\ d e. RR+ /\ 0 < ( F ` d ) ) ) -> 0 < B ) |
220 |
219
|
3anassrs |
|- ( ( ( ( ph /\ B e. RR+ ) /\ d e. RR+ ) /\ 0 < ( F ` d ) ) -> 0 < B ) |
221 |
90 44 7
|
mul2lt0rgt0 |
|- ( ( ph /\ 0 < B ) -> A < 0 ) |
222 |
217 220 221
|
syl2anc |
|- ( ( ( ( ph /\ B e. RR+ ) /\ d e. RR+ ) /\ 0 < ( F ` d ) ) -> A < 0 ) |
223 |
4 222
|
eqbrtrrid |
|- ( ( ( ( ph /\ B e. RR+ ) /\ d e. RR+ ) /\ 0 < ( F ` d ) ) -> ( C ` 0 ) < 0 ) |
224 |
216 223
|
eqbrtrd |
|- ( ( ( ( ph /\ B e. RR+ ) /\ d e. RR+ ) /\ 0 < ( F ` d ) ) -> ( F ` 0 ) < 0 ) |
225 |
|
simpr |
|- ( ( ( ( ph /\ B e. RR+ ) /\ d e. RR+ ) /\ 0 < ( F ` d ) ) -> 0 < ( F ` d ) ) |
226 |
224 225
|
jca |
|- ( ( ( ( ph /\ B e. RR+ ) /\ d e. RR+ ) /\ 0 < ( F ` d ) ) -> ( ( F ` 0 ) < 0 /\ 0 < ( F ` d ) ) ) |
227 |
206 208 206 209 211 212 215 226
|
ivth |
|- ( ( ( ( ph /\ B e. RR+ ) /\ d e. RR+ ) /\ 0 < ( F ` d ) ) -> E. z e. ( 0 (,) d ) ( F ` z ) = 0 ) |
228 |
207 109 110
|
3syl |
|- ( ( ( ( ph /\ B e. RR+ ) /\ d e. RR+ ) /\ 0 < ( F ` d ) ) -> ( E. z e. ( 0 (,) d ) ( F ` z ) = 0 -> E. z e. RR+ ( F ` z ) = 0 ) ) |
229 |
227 228
|
mpd |
|- ( ( ( ( ph /\ B e. RR+ ) /\ d e. RR+ ) /\ 0 < ( F ` d ) ) -> E. z e. RR+ ( F ` z ) = 0 ) |
230 |
205 229
|
syldan |
|- ( ( ( ( ph /\ B e. RR+ ) /\ d e. RR+ ) /\ A. f e. RR+ ( d <_ f -> ( abs ` ( ( ( F ` f ) / ( f ^ D ) ) - B ) ) < B ) ) -> E. z e. RR+ ( F ` z ) = 0 ) |
231 |
166
|
adantr |
|- ( ( ph /\ B e. RR+ ) -> A. e e. RR+ E. d e. RR+ A. f e. RR+ ( d <_ f -> ( abs ` ( ( ( F ` f ) / ( f ^ D ) ) - B ) ) < e ) ) |
232 |
|
simpr |
|- ( ( ph /\ B e. RR+ ) -> B e. RR+ ) |
233 |
|
simpr |
|- ( ( ( ph /\ B e. RR+ ) /\ e = B ) -> e = B ) |
234 |
233
|
breq2d |
|- ( ( ( ph /\ B e. RR+ ) /\ e = B ) -> ( ( abs ` ( ( ( F ` f ) / ( f ^ D ) ) - B ) ) < e <-> ( abs ` ( ( ( F ` f ) / ( f ^ D ) ) - B ) ) < B ) ) |
235 |
234
|
imbi2d |
|- ( ( ( ph /\ B e. RR+ ) /\ e = B ) -> ( ( d <_ f -> ( abs ` ( ( ( F ` f ) / ( f ^ D ) ) - B ) ) < e ) <-> ( d <_ f -> ( abs ` ( ( ( F ` f ) / ( f ^ D ) ) - B ) ) < B ) ) ) |
236 |
235
|
rexralbidv |
|- ( ( ( ph /\ B e. RR+ ) /\ e = B ) -> ( E. d e. RR+ A. f e. RR+ ( d <_ f -> ( abs ` ( ( ( F ` f ) / ( f ^ D ) ) - B ) ) < e ) <-> E. d e. RR+ A. f e. RR+ ( d <_ f -> ( abs ` ( ( ( F ` f ) / ( f ^ D ) ) - B ) ) < B ) ) ) |
237 |
232 236
|
rspcdv |
|- ( ( ph /\ B e. RR+ ) -> ( A. e e. RR+ E. d e. RR+ A. f e. RR+ ( d <_ f -> ( abs ` ( ( ( F ` f ) / ( f ^ D ) ) - B ) ) < e ) -> E. d e. RR+ A. f e. RR+ ( d <_ f -> ( abs ` ( ( ( F ` f ) / ( f ^ D ) ) - B ) ) < B ) ) ) |
238 |
231 237
|
mpd |
|- ( ( ph /\ B e. RR+ ) -> E. d e. RR+ A. f e. RR+ ( d <_ f -> ( abs ` ( ( ( F ` f ) / ( f ^ D ) ) - B ) ) < B ) ) |
239 |
230 238
|
r19.29a |
|- ( ( ph /\ B e. RR+ ) -> E. z e. RR+ ( F ` z ) = 0 ) |
240 |
1 2
|
dgreq0 |
|- ( F e. ( Poly ` RR ) -> ( F = 0p <-> ( C ` D ) = 0 ) ) |
241 |
5 240
|
syl |
|- ( ph -> ( F = 0p <-> ( C ` D ) = 0 ) ) |
242 |
241
|
necon3bid |
|- ( ph -> ( F =/= 0p <-> ( C ` D ) =/= 0 ) ) |
243 |
6 242
|
mpbid |
|- ( ph -> ( C ` D ) =/= 0 ) |
244 |
3
|
neeq1i |
|- ( B =/= 0 <-> ( C ` D ) =/= 0 ) |
245 |
243 244
|
sylibr |
|- ( ph -> B =/= 0 ) |
246 |
|
rpneg |
|- ( ( B e. RR /\ B =/= 0 ) -> ( B e. RR+ <-> -. -u B e. RR+ ) ) |
247 |
246
|
biimprd |
|- ( ( B e. RR /\ B =/= 0 ) -> ( -. -u B e. RR+ -> B e. RR+ ) ) |
248 |
247
|
orrd |
|- ( ( B e. RR /\ B =/= 0 ) -> ( -u B e. RR+ \/ B e. RR+ ) ) |
249 |
44 245 248
|
syl2anc |
|- ( ph -> ( -u B e. RR+ \/ B e. RR+ ) ) |
250 |
174 239 249
|
mpjaodan |
|- ( ph -> E. z e. RR+ ( F ` z ) = 0 ) |